Instructions

On the first page of your solution write-up, you must make explicit which problems are to be graded for "regular credit", which problems are to be graded for "extra credit", and which problems you did not attempt. Please use a table something like the following

Problem

01

02

03

04

05

06

07

08

09

...

Credit

RC

RC

RC

EC

RC

RC

NA

RC

RC

...

where "RC" is "regular credit", "EC" is "extra credit", and "NA" is "not applicable" (not attempted). Failure to do so will result in an arbitrary set of problems being graded for regular credit, no problems being graded for extra credit, and a five percent penalty assessment.

You must also write down with whom you worked on the assignment. If this changes from problem to problem, then you should write down this information separately with each problem.

Specific Instructions

In Problems 1, 2, and 3, you must explain how one type
of machine can simulate one or more other machines. (For example,
in part of Problem 1, you must show how a TM can simulate a
2-PDA; in Problems 2, you must show how one TM can simulate
two other TMs.) You may describe these simulations at a high
level, in English, as we did in class.

In Problem 4, you should provide a diagonalization proof, as we did in class
and as presented in the text. You must show, by contradiction using diagonlization,
that no one-to-one mapping can exist between the infinite sets in question.

Problems

Required: 3 of the following 4 problemsPoints: 30 pts per problem

Exercise 3.9

Hint: For part (a), show that 2-PDAs can simulate
Turing Machines, and for part (b), show that Turing Machines
can simulate 3-PDAs. (For the latter part, you will likely need
multi-tape Turing Machines, which can themselves be simulated
by ordinary Turing Machines.)

Exercise 3.15 (d,e)

Hint: You may use multi-tape Turing Machines which we have proven
to be equivalent in power to ordinary Turing Machines.

Exercise 3.15 (b,c)

Hint: You may use multi-tape Turing Machines which we have proven
to be equivalent in power to ordinary Turing Machines.

Let N be the set of natural numbers; i.e.,
N = {1, 2, 3, ...}. Prove, by diagonalization, that the
cardinality of the power set of N is greater than the
cardinality of N; i.e.,
|2N| > |N|.

(Hint: Let X be the ordered set {a,
d, k, z}. Then one can denote the
subsetY = {a, k, z}
by the sequence 1011; each "1" indicating that the corresponding
element from the original set is present in the subset, and each "0"
indicating that the corresponding element is absent.)